This is the math problem I've been hitting my head against the wall over. I can visualize it, and think I have come up with the correct formula, but the numbers I end up with won't crunch. So, for your reading pleasure, and hopefully to find someone who can explain this to me in a way that works...
OK, the problem goes like this:
You have a rectangle of cardboard, 40" long and 20" wide. You want to make a pizza box from it. You will have to cut a square from each corner, and the same sized square from the middle of the long side, in order to fold it in half and have a lid as well as a bottom.
What size squares do you cut in order to have the maximum amount of volume in the box?
So you can visualize it, this would be it, folding along the perforated edges for the sides & lid:
Given the formula to find volume is LWH, the formula should be:
[(40-3H)/2] x (20-2H) x H
Just from trial and error I think the answer will be somewhere around 4.5, but I can't seem to get that solution.
What I get is:
[(40-3H)/2](20-2H)(H) =
(800-80H-60H+6H^2)H =
800H-140H^2+6H^3 - putting it into standard format:
6H^3 - 140H^2 + 800H - factoring out 2H
(3H^2 - 70H + 400) - plugging it into the quadratic formula:
H = 70 +/- sqrt70^2 - 4(3)(400) divided by 2(3) - giving:
70+/- sqrt4900-4800 divided by 6 = 70+/-10 divided by 6
This results in either 13.3333 or 10
OK, What am I doing wrong?????